Both the two mentioned approaches have some valuable aspects together with critical points to be underlined. Projection techniques are strongly connected with physical laws of the problem since they use modal basis functions obtained by real solutions to extract the main dynamics and they employ those modes to project and reconstruct conservation equations solutions manifolds. Unfortunately non-linearity and non-affinity of the parametrized formulation can be difficult issues to be carefully treated. Moreover sometimes it may happen that the equations are not directly available and in that case these methods are not employable. A classical example is constituted by commercial software where a deep description of the employed laws is not provided. Again, in the literature one may find many examples of very slow decay of the eigenvalues for specific problems, that means many modal basis functions are needed to reach a decent approximation for the solutions, wasting, de facto, the whole possible gain coming from such an architecture ([1]}). A very well known problem affected by such an issue is represented by turbulent flows (see e.g. [2]}, [3]}, [4]}, [5]}, [6]}). On the contrary Artificial Intelligence (AI) techniques are very versatile. They just need a set of solutions to be trained and, despite the complexity of the mathematical formulation of the starting problem, they can be constructed and modeled to pursue good approximations in a short time. The dark side of these approaches is the fact that they have a much weaker connection with the real physics of what they are approximating and the actual meaning of every single part constituting their architecture is arguably comprehensible in terms of phenomena representation. For this reason they may give inaccurate results thanks to impossibility in having a deeper check on networks responses.
